1. Field of the Invention
This application relates to mathematical modeling of pipelines to transport hydrocarbons and other products with volatile components.
2. Background of the Related Art
Pipelines are used to transport liquid hydrocarbon products to ports, processing facilities or refineries. Pumps move the product through the pipeline. Pump stations are deployed at intervals along the pipeline route to compensate for friction head loss and to move product from lower elevations up to and beyond peak elevations. Conditions along the pipeline are monitored using sensors to detect pressure, temperature, flow rate and other measurable properties. Transmitters coupled to the sensors send data to a central processor that uses the data to help drive a model of fluid flow between sensors. One of the key factors in determining the flow is the head, which is related to the pressure, elevation and density of the fluid.
Factors affecting the hydraulic head at any given point along the pipeline include friction head loss and elevation. Friction head loss occurs when product pumped through a pipeline physically interacts with an interior wall of the pipeline and with pipeline features such as expansion loops, bends and valves that may impair flow of product through the pipeline. Friction head loss generally increases with increased flow rates and/or with increased product viscosity, and accumulates with distance along the pipeline downstream of a pump station. Friction head loss per length of pipeline is generally constant for a pipeline of constant diameter and at a steady flow rate. Computer programs may be used to model pipeline flow between sensors by solving the equations of motion which include simulating friction head loss and elevation head.
An elevation profile indicates the elevation of the pipeline at all points along the pipeline route, and indicates the head required to move product through the pipeline. For example, higher elevations downstream of a pump require greater pump discharge head because the product must be moved up to and over peak elevations against the force of gravity.
In purely liquid-phase flow, called “tight” flow, the physics is well understood and is often written as equations of conservation of mass, momentum and energy as well as the physical properties of the fluid being transported, typically referred to as equations of state. Computer modeling of tight flow regions of pipelines is well known in the art.
Products transported by pipelines, such as oil, are often volatile. The vapor pressure of a volatile product generally depends on the chemical composition and on the propensity of lighter components in the product to evaporate to form a vapor phase within the pipeline. For example, oil may comprise a range of hydrocarbons including pentane (C5H12), hexane (C6H14), heptane (C7H16) and octane (C8H18). The lighter (lower molecular weight) components (e.g., pentane and hexane) will, at a pressure below the vapor pressure, evaporate to form a vapor phase within the pipeline. Evaporation of lighter components of volatile products is promoted by low pressures (below the vapor pressure) and high temperatures.
A vapor phase within a pipeline generally resides above the remaining (un-evaporated) liquid which will, in the presence of a vapor phase, flow through the pipeline with a free surface in contact with the vapor phase. This is known as open channel flow (OCF) or, in the pipeline industry, as slack flow. A slack region of a pipeline is a portion of the pipeline where OCF or slack flow occurs. The classical example of OCF is a river flowing through a river bed. In OCF, the liquid phase flows downstream through the pipeline under the influence of gravity and under generally constant pressure. In a river bed, the pressure at the surface of the liquid phase is equal to atmospheric pressure. In a closed pipeline, however, the pressure at the surface of the liquid phase in a slack region is equal to the vapor pressure of the volatile component. The mass of the volatile component (vapor phase) contained within a finite section of the pipeline, being gaseous and at low pressure, is negligible compared to the mass of the liquid component within that same finite section. Slack flow generally begins within a pipeline at points downstream of peak elevations where the head is lower than the vapor pressure (due, for example, to a high elevation profile), and slack flow continues downslope until the pressure within the pipeline again rises above the vapor pressure of the product. It is also possible, although rare, for slack conditions to begin where the downward slope increases, without actually having a local maximum elevation. Sections of the pipeline where slack flow is occurring are referred to as slack regions.
Because the pressure at the surface of the liquid phase is constrained to be equal to vapor pressure in slack regions, the standard “tight line” version of the equations of motion do not adequately describe the behavior of product flowing through slack regions of the pipeline. Hence, conventional pipeline flow models accurately predict pipeline flow in tight flow regions but are insufficient for modeling pipeline flow in slack regions. In tight flow regions, pipeline head decreases with increased elevation and increases with decreased elevation, and the magnitude of change in head for a given change in elevation depends on the density of the product. The pump discharge head needed to move product up to and over a peak elevation depends on the elevation of the peak, the density of the product and the friction head loss from the pump to the peak elevation. The substantially incompressible liquid phase in tight flow regions provides a direct correlation between elevation and head. Slack flow, by contrast, allows product to move through elevation changes without change in pressure and independently of changes occurring in downstream regions of the pipeline.
FIG. 1 is a graph 10 illustrating relationships among the maximum operating pressure (MOP) profile 12 of a pipeline, the actual pipeline head 14 and the elevation profile 16 of the terrain traversed by the pipeline. The pipeline head 14 must remain below the MOP profile 12 to prevent from damaging the pipeline or exceeding a safe operating pressure. To maintain tight line flow, the pipeline head 14 must remain above the elevation profile 16. However, it may not be possible or economical to provide sufficient head to remain above peak elevations indicated at points B and D without unwanted encroachment upon the MOP profile 12. An interval or band formed between the MOP profile 12 and the elevation profile 16 is the targeted operating zone. As long as pipeline head 14 remains within this zone, the product will move through the pipeline without damage to the pipeline.
The hydraulic head profile 14 generally lies along a straight line from point A to point F, but there are features of the hydraulic head profile 14 that reveal the effects of friction and slack flow. For example, line segments A-B, C-D and E-F exhibit a generally common, constant, and negative slope corresponding to the friction head loss per unit distance for a given product flowing at a given product flow rate. A different product and/or a different flow rate will generally exhibit a different slope in these same portions of the pipeline because the amount of friction head loss per unit distance would be different. Deviations from the generally linear decline in head occur at line segments B-C and at D-E as a result of slack flow. Slack flow causes the hydraulic head 14 to track along and equal the elevation profile 16 until tight flow is restored.
Modeling in slack flow regions adds complexity. Pressure becomes decoupled from frictional loss and instead is pegged to the vapor pressure of the product. The model should be adapted to calculate and account for the reduced liquid cross-sectional flow area within such slack flow regions. None of these factors are present in tight flow regions that conform to conventional tight-line models. The liquid phase of the product flows in open channel flow in slack flow regions of the pipeline. Under steady state conditions, the cross-sectional area occupied by the vapor phase of the product in a slack flow region of the pipeline depends solely on the elevation and the slope, but certain changes in the product flow will cause the slack region to expand or contract in cross-section as well as to longitudinally retract or extend along regions of the pipeline.
FIG. 2 is an elevation section view of a descending portion 20 of a pipeline having a diameter 25. The vapor phase 22 meets the liquid phase 24 along an interface 23, and the vapor phase 22 and the liquid phase 24 of the product co-exist with the liquid phase 24 moving in OCF, meaning that the head is equal to the vapor pressure of the product and the descending portion 20 is in a slack flow region of the pipeline. The liquid phase 24 moves along the descending portion 20 of the pipeline in the direction of arrow 26. A discrete mathematical model of the pipeline considers the pipeline as a contiguous set of “computational cells” over which reasonable approximations to the continuum equations describing the flow of product within the pipeline can be written. The primitive variables to be tracked and approximated will be prescribed to the upstream and downstream ends of the cell, and these upstream and downstream cell boundaries will be referred to henceforth as “knots.”
A model relies on using a series of determinations of the conditions existing at different times within a common set of computational cells. A model having this structure takes into account that there may be changes over time that occur in one or more of the factors that determine the conditions existing within any given computational cell in a pipeline. The term “time step,” as that term is used herein, means the difference in time from a determination of the conditions within a computational cell to a subsequent determination of the conditions within the computational cell.
A computational cell 21 as illustrated in FIG. 2, referred to herein below as a “cell,” may lie, for example, within the descending portion 20 as defined between upstream (and upslope) knot 28 and downstream (and downslope) knot 29. It will be understood that knot 28 is also the downstream knot of an upstream cell (not fully shown in FIG. 2) adjacent to cell 21 and knot 29 is the upstream knot of a downstream cell (not fully shown in FIG. 2) adjacent to cell 21.
Pipelines are dynamic, and changes in flow rates, elevation and physical properties of the product moving through the pipeline can cause tight flow regions to change to slack flow regions and slack flow regions to change to tight flow regions. For example, in one instance, rapid transitioning from tight flow to slack flow may occur along gently descending portions of the pipeline at elevations that cause the head to be generally equal to the vapor pressure of the product, just as rapid transitioning from slack flow mode to tight flow mode may occur along gently ascending portions at the same elevations. In another instance, rapid transitioning from tight flow to slack flow may occur along both the ascending and descending portions of a pipeline straddling a peak elevation where a product pump is shut down, thereby causing depressurization of the pipeline both upstream of and downstream of the peak elevation. In yet another instance, rapid transitioning from slack flow to tight flow may occur along portion of a pipeline downstream of a pipeline junction where a product provided from a supplemental pump is suddenly introduced into the pipeline, or rapid transitioning from tight flow to slack flow may occur along this same portion of the pipeline where the supplemental pump is suddenly deactivated or isolated from the pipeline. In another instance, rapid transitioning from tight flow to slack flow may occur along portions of a pipeline upstream of an off take when the flow rate at the off take is increased, or rapid transitioning from slack flow to tight flow may occur along this same portion of the pipeline when the flow rate at an off take is decreased. Under these and other circumstances, multiple cells within the pipeline being modeled can change from being tight to being slack in a single time step. A change from tight to slack, or vice versa, is referred to a change in flow “mode.” Conventional tight-line pipeline flow models produce non-physical results (e.g., calculates an impossible negative pressure) under these conditions. Similarly, conventional slack line models can become computationally unstable under these circumstances. The result is compromised surveillance in transitioning regions of the pipeline and a corresponding loss of leak detection capacity, product batch tracking capability, and loss of other important pipeline surveillance and control functions that depend on accurate modeling of the pipeline.
FIG. 3 is a cross-section view of the cell 21 within the descending portion 20 of FIG. 2, again illustrating the dual phase nature of product flowing in a slack flow mode. As in FIG. 2, FIG. 3 illustrates the vapor phase 22 and the liquid phase 24 co-exist in open channel flow, with an interface 23 there between. It should be noted that the cross-sectional flow area of the liquid phase 24 is calculated using the angle 27 formed between a first radius (directed from the center of the pipe diameter 25 to an endpoint of the phase interface 23) and a second, vertical radius (directed from the center of the pipe diameter 25 to the bottom of the descending portion 20 of the pipeline). Using θ for the angle 27 (in radians) formed between a first radius and the second, vertical radius, and using D for the diameter 25 of the interior of the pipeline, the equation for the cross-sectional flow area, A, of the liquid phase 24 in FIG. 3 is:
  A  =                    D        2            4        ⁢          (              θ        -                              1            2                    ⁢          sin          ⁢                                          ⁢          2          ⁢                                          ⁢          θ                    )      
The liquid flow cross-sectional area is used to model conditions of state at each cell not in tight flow mode. It will be noted that tight flow requires that θ=π and, therefore, the cross-sectional flow area of the liquid phase 24 in tight flow regions is simply the cross-sectional area of the descending portion 20 of the pipeline. It should be understood that these concepts apply equally to ascending or horizontal portions of the pipeline where slack flow may also occur.
Transitions from slack flow to tight flow, and transitions from tight flow to slack flow, occur under circumstances other than sloping (ascending or descending) portions of the pipeline at high elevations. Other circumstances that cause these transitions to occur include liquid product drain-down on both sides of an elevation peak, start-up and/or shut-down of a pump discharging into the pipeline, or the opening or closure of a valve hydraulically connected to the pipeline. FIGS. 4-6 illustrate these circumstances where conventional models yield poor modeling results.
FIG. 4 depicts elevated terrain 37 traversed by a pipeline segment 30 disposed, for example, between an upstream pressure sensor 42 and an upstream temperature sensor 43 on a first side and a downstream pressure sensor 40 and a downstream temperature sensor 41 on a second side, the pipeline segment 30 comprising a peak elevation 35 disposed between an ascending portion 31 and a descending portion 32. A pump 81 is disposed upstream of the ascending portion 31 to move product received from a source 45 up the ascending portion 31 and beyond the peak elevation 35. Upon shutdown of the pump station 81, the hydraulic head throughout the pipeline segment 30 downstream of the pump station 81 will decrease. In some cells where the head falls below the vapor pressure of the product, a transition from tight flow to slack flow will occur in the pipeline segment 30. As the slack region expands, the slack region will migrate upslope in both the ascending portion 31 and the descending portion 32 of the pipeline segment 30 and the liquid will drain downslope in both the ascending portion 31 and the descending portion 32 and in the direction of arrows 33 and 34, respectively. It will be understood that under certain conditions the shut down of pump 81 causes a range of cells within the pipeline segment 30 to transition from tight flow to slack flow.
FIG. 5 illustrates an ascending portion 31 of a pipeline segment in fluid communication with a first pump 82 receiving product from a first source 46 and a second pump 83 receiving product from a second source 47, the second pump 83 being isolatable from the ascending portion 31 of the pipeline by closure of a valve 54. Product flow is along the direction of arrow 38. Upon start-up of the first pump 82 or the second pump 83 (with valve 54 open), or upon opening of valve 54 (with second pump 83 active), a positive pressure surge will move through the ascending portion 31 of the pipeline raising the hydraulic head and, within a range of cells with head at the vapor pressure of the product, transition from slack flow to tight flow mode will occur. Similarly, upon shut-down of the first pump 82 or second pump 83 (with valve 54 open), or upon closure of valve 54 (with second pump 83 active), a negative pressure surge propagates through the ascending portion 31 of the pipeline lowering hydraulic head and, within a range of cells with head at the vapor pressure of the product, transition from tight flow mode to slack flow mode will occur. A pressure sensor 48 and a temperature sensor 49 may be disposed on a first pipeline feed lateral 52 to provide data relating to the conditions of the product within the first pipeline feed lateral 52, and a pressure sensor 50 and a temperature sensor 51 may be disposed on a second pipeline feed lateral 53 to provide data relating to the conditions of the product within the second pipeline feed lateral 53.
FIG. 6 illustrates a descending portion 32 of a pipeline in fluid communication with a first storage tank 90 and a second storage tank 91, the first tank 90 and second tank 91 being isolatable from the descending portion 32 of the pipeline using the first valve 58 and/or second valve 57, respectively. A first tank discharge pipe 92 and a second tank discharge pipe 93 may be provided to drain the first tank 90 and the second tank 91. Product flow is along the direction of arrow 39. A pressure sensor 94 may be provided near the junction of the first tank feed pipe 55 and the second tank feed pipe 56 to provide data relating to the conditions of the product at or near the junction. Upon opening of the first valve 58 or second valve 57, or both, a negative pressure surge propagates up the descending portion 32 and, in a range of cells with head just above the vapor pressure of the product, a transition from tight flow mode to slack flow mode will occur. Upon closure of the first valve 58 or the second valve 57, or both, a positive pressure surge propagates up the descending portion 32 of the pipeline and, in a range of cells with head at the vapor pressure, transition from slack flow to tight flow mode will occur.
It will be understood that these and other changes in pipeline operating modes can cause transitions between tight flow mode and slack flow mode to occur and it is important that these transitions be accurately modeled in order to sustain certain pipeline surveillance and control capabilities. It is feasible to construct a pipeline with a greater number of pumps or robust pumps having greater discharge head and with higher rated valves and pipe materials in order to maintain the product in the liquid phase at all points along the pipeline route. However, such a robust pipeline would be prohibitively expensive to build due to the great cost of the pumps, valves and pipe, and due to the increased construction cost, and it would be prohibitively expensive to operate due to the increased requirement for pump drivers and input power. The product delivered to a port, tanker or refinery by such a robust pipeline system would have the same value as the product shipped through a conventional pipeline that tolerates the formation of slack regions at remote, higher elevations.
It is therefore advantageous to tolerate and manage the formation of a slack region within the pipeline in order to dramatically reduce both the cost of the pipeline system and the cost of operating the pipeline. There is a need for a method of modeling pipeline conditions in: regions affected by changes from tight flow mode to slack flow or from slack flow mode to tight flow; zero flow (shut-in) conditions in slack flow regions; transitions from flowing conditions to shut-in conditions in slack flow regions; transitions from shut-in conditions to flowing conditions in slack flow regions; transitions from forwards flow conditions to backwards flow conditions in slack flow regions; transitions from backwards flow conditions to forwards flow conditions in slack flow regions; and drain down conditions on both sides of an elevation peak.